Note that a Hausdorff space, according to this definition, needn't be a topological space, and vice versa.

Call the elements of $\tau$ "open sets", whether or not $\tau$ is a topology.

Then many of the definitions of topological spaces apply equally well to Hausdorff spaces. In particular, the notion of a convergent sequence can be defined as per usual.

In particular, define that a sequence $x : \mathbb{N} \rightarrow X$ converges iff there exists $a \in X$ such that for all open $A$ such that $a \in A$, there exists $N$ such that for all $n \geq N$ it holds that $x_n \in A$.

Then we can write this proof to show that every convergent sequence in $X$ has a unique limit. (i.e. $a$ is unique).

Presumably, the more general result that every net has a unique limit can also be proven.

So why not study limits in arbitrary Hausdorff spaces? What goes wrong when we neglect to assume statements like, "The arbitrary union of open sets is open"?

Edit: So to clarify, my interest lies in the interaction between the topological space axioms and the Hausdorff axiom. What is it about a Hausdorff topological space that is so magical? There must be some sort of synergy going on, or they would have been studied independently of one another.

Note that we can define continuity of functions between topological spaces in terms of preimages of open sets. Lets call this "pre-images continuous." And we can define continuity of functions between Hausdorff spaces in terms of limits of nets. Lets call this "limit-continuous." Perhaps these notions coincide precisely in the case of Hausdorff topological spaces?

Note also that the topology generated by a Hausdorff space is necessarily Hausdorff. So for any Hausdorff space $X$, lets $X'$ denote the (necessarily Hausdorff) topological space generated by $X$. Perhaps a function $f : X \rightarrow Y$ between Hausdorff spaces is limit-continuous if and only if $f : X' \rightarrow Y'$ is (open-preimages)-continuous.

1 Answer
1

It has not yet come up naturally in work on theories / problems that we currently study.

Perhaps a more blunt phrasing of the second answer would be

2'. It has not yet come up in a situation where anyone would care.

There is also sometimes a third answer, which is

3. People do study that, it's just not mainstream.

For your particular question, about spaces only assumed to satisfy the Hausdorff separation axiom:

I haven't really put much effort in figuring out whether answer 1 applies, but I think it's quite likely that answer 2 does. Maybe it's true that an intricate theory of these Hausdorff-only-spaces could be developed; but as far as I'm aware, currently, there is no reason anyone would care. I could just as easily ask about the theory of "droops", which are (let's say) sets equipped with a ternary operation $\oslash$ satisfying the identity
$$\oslash(a,\oslash(a,b,c),\oslash(c,b,a))=\oslash(b,a,c).$$
Unless, and until, someone demonstrates that such a thing occurs naturally in the mathematics we're already doing, I find it very hard to be interested; and indeed, the theory of these things themselves would suffer, due to a lack of intuition and interconnection with the rest of mathematics. Once the world realizes that the set of complex structures on a manifold, or the rational points on an elliptic curve, naturally forms a droop, then you'll start getting people interested in droops, and moreover, this connection will give you things to think about, things to ask, things to investigate, and the theory of droops will begin to flourish. Without such a connection, any mathematician who was polite and listened to someone explain the definition of a droop most likely wouldn't be able to say anything about them because they lack any previous experience or intuition for such an object.

Okay, I'm exaggerating a bit with droops; Hausdorff-only-spaces are somewhat closer to everyday mathematical experience than a droop. And I don't mean to sound angry or discouraging; it's great to ask questions about why we do things one way and not the other, especially when answer 1 applies and the ways in which the general case are worse have been thoroughly investigated, and you'll gain some real mathematical experience from learning about why we're doing things the way we are. But the overall point is hopefully clear.